Paradox Lost
Paradox Lost: The Evolution of Strategies in Selten's Chain Store Game
William M. Tracy1
Rensselaer Polytechnic Institute
ABSTRACT
The classical game theoretic resolution of Selten's Chain Store Game is unsatisfactory; it alters
the game to avoid the paradox, yet still fails to organize the existing experimental data. When
co-evolutionary algorithms are applied to the Chain Store game, the resulting system's dynamics
are neither intuitively paradoxical, nor contradicted by the existing experimental data. This has
implications for real world entrant-incumbent strategy decisions. The role of strategy
deterioration, or genetic drift, in facilitating the system's dynamics is also noteworthy. The
efficacy of co-evolutionary algorithms in resolving the Chain Store Paradox supports their
potential to improve strategic modeling.
JEL: C73, C72
Keywords:
market entry, computation, genetic drift, equilibria selection, Chain Store Paradox
1
The author has benefited immensely from conversations and correspondences with: John Miller, Bill McKelvey,
Hans Schollhammer, Phil Bonacich, Sandy Jacoby, John Holland, Herb Gintis, Lee Alternberg, Jan Rivkin, Nick
Gessler, Elise Hui, Scott Carr, HAN Jing, Bryan Routledge, Jeremy Avnet, Allan Freidman, Hiroshi Tamura,
ZHANG Zhi, and ZHU Mengxiao. The author assumes sole responsibility for any mistakes or inaccuracies in this
paper.
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1. INTRODUCTION
The Chain Store Game is typically described in terms of geographic markets and local
competitors. However, the strategic issues that underlie the game also speak directly to entrantincumbent decisions in product markets. In addition to a Wal-mart or Carrefour, the insights of
this game are also applicable to firms like Microsoft. Microsoft enjoys an incumbent, dominant
position in a finite number of product markets, each of which can be credibly challenged by only
a finite number of potential entrants.
The orthodox resolution to Selten's Chain Store Paradox, involves changing the game
through the addition of alternative 'types' of incumbent firms. This is intuitively appealing when
one imagines the Chain store to be an anonymous supermarket. However, there are some
situations in which the typing solution is less appealing. An incumbent such as Microsoft is
unlikely to be confused with a different 'type' of Microsoft. As Bill Gates' personal role recedes,
and Microsoft is strategic decisions are increasingly influenced by the expectations of financial
markets, the intuitive appeal of typing further decreases.
The inability of orthodox game theory to provide a non-paradoxical resolution to Selten's
specification of the Chain Store Game, is also troubling for practitioners who use game theory to
assess the merits of real world strategic decisions. Under Selten's specification of the Chain Store
game, the limitations of the game theoretic analysis are obvious. This raises the possibility of
other situations in which similar limitations exit, but are less obvious. Indeed, there is a wealth of
experimental game theoretic work that reveal a disconnect between human behavior and game
theory's predictions (examples include: McKelvey and Palfrey 1992; Hoffman, McCabe, and
Smith 1996; McCabe and Smith 2000; Colman 2003). More troubling is the experimental data,
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which show the large impact of culture and society on game behavior (Henrich and Smith 2003,
Hill and Gurvan 2003, Tracer 2003, and Alvaed 2003).
The above is not intended to understate the important insights and understanding
orthodox game theory has generated. However, it would be useful to identify a new solution
concept, which both preserves the important findings of orthodox game theory, and resolves
some of these contradictions. This suggests a reexamination of game theory's core assumptions.
Game theory typically assumes that players' (near) perfect rationality and playoffs are
Common Knowledge. However, real-world firms seldom implement the analytical solution
concept dictated by these assumptions. Firms do not attempt to locate a strategy that constitutes
a fixed point in a game's joint-strategy space. Rather, firms typically observe and emulate the
strategies of successful peers. Before an equilibrium is reached, this emulation-based adjustment
process might generate dynamic, and seemly stochastic, firm behavior. Behavior during this
dynamic period could easily influence which, if any, equilibrium the system eventually selects.
Firms' dynamic, emulation-learning processes frequently involve modifying or
combining the strategies of successful peers. These behaviors are similar to the selection,
mutation and crossover operators employed in evolutionary algorithms. Therefore, a strategic
solution concept employing computational evolution might refine our understanding of
equilibrium selection and transition. Indeed, there is a growing literature using evolutionary
computational learning models to evolve solutions to game theoretic constructions (e.g. Axelrod
1984; Axelrod 1987; Miller 1988; Holland and Miller 1991; Young 1993; Andreoni and Miller
1995; Young 1996; David 1999; Ünver 2001; Arifovic 2001; Hart 2002; Haruvy, Roth and
Ünver 2006).
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The validity of this technique is still controversial. To support the usefulness of
evolutionary computation, this paper applies a computational evolutionary algorithm to Selten's
Chain Store Game. This paper examines classes of evolutionary algorithms whose solution to the
chain store game is neither paradoxical nor contradicted by existing behavioral data. In
particular, the solution suggests that entrants should expect incumbents to fight price wars more
frequently than predicted by orthodox game theory.
This paper's evolutionary system's behavior approximates an ergodic Markov process.
Each of the game's Nash Equilibria (NE) are highly absorptive states. Transitions between NE
are driven by a phenomenon akin to genetic drift. Although no state is perfectly absorptive, the
game's singular sub-game perfect equilibrium is the most absorptive. This modestly suggests that
we can use evolutionary computation as a solution concept that enhances our understanding of
disequilibrium behavior and equilibrium selection, while retaining many of the equilibrium
identifications and rankings supplied by classical game theory.
The next section of this paper reviews Selten's Chain Store game, and examines why
classical game theory's orthodox resolution is unsatisfactory. The third section introduces the
concept of evolutionary computation and formalizes this paper's hypotheses. The fourth section
details a class of evolutionary algorithms whose solution to the Chain Store game better explains
observed behavior. The fifth section examines the simulation results and uses concepts from
evolutionary theory to examine how genetic drift causes the system to diverge from the game
theoretic solution. The sixth section concludes by noting implications for strategic analysis and
evolutionary computation.
2. THE CHAIN STORE PARADOX
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2.1 The Chain Store Game
Reinhard Selten proposed the follow stylization of the strategic interaction between a
chain store and potential entrants.2 Let Ѓ be an n-stage, (n+1)-player game. Call the (n+1)th
player "Chain Store", and call the remaining n players: First Entrant, Second Entrant, … nth
Entrant. At the start of the ith stage, the ith Entrant observes the outcome of the previous i-1
stages and then decides to either 'enter' or 'stay-out' of the market. If the Entrant chooses to 'stayout,' the Chain Store gets a payoff of 'A' and the Entrant gets a payoff of 'C.' If the Entrant
chooses to enter the market, the Chain Store must choose to either 'fight' or 'acquiesce.' If the
Chain Store chooses to 'fight,' both the Chain Store and the Entrant gets a payoff of 'D.' If the
Chain Store chooses to "acquiesce," both the Chain Store and the Entrant gets a payoff of 'B.' By
construction, A > B > C > D. Table 1 summarizes the payoffs from a given stage.
---------------------Insert Table 1 about here
---------------------Each Entrant only plays one stage of the game, so Table 1 depicts the total payoff for the
ith Entrant. The Chain Store's total payoff is the sum of its payoffs from each of the n stages.
Each player has a unique strategy space. Let Sc represent the Chain Store's strategy space
and Si represent the ith Entrant's strategy space. The First Entrant's strategy space, S1, contains
only two elements; 'Enter' or 'Stay-Out.' However, the Second Entrant can condition its decision
to 'Enter' or 'Stay-Out' based on the outcome of the first stage. There are three possible first
stage outcomes; Enter-Acquiesce, Enter-Fight, and Stay-Out. Hence, S2 contains 23 elements.
More generally, there are 3^(i-1) possible outcomes that the ith Entrant may observe. Therefore
the ith Entrant has 2^(3^(i-1)) elements in its strategy space. Following similar logic, the Chain
N
Store has ∑ 2^(3^(i-1)) elements in its strategy space. Figure 1 depicts the extensive form of a
i=1
2
Selten (1978)
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two-stage version of this game using Selten's payoffs in which: A = 5, B = 2, C = 1, and D = 0.
---------------------Insert Figure 1 about here
--------------------2.2 The Game Theoretic Solution
Selten sets n equal to twenty and attempts to solve the game using backward induction.
Selten observes that the Chain Store's payoff from the first nineteen stages should not influence
its behavior in the twentieth stage. If the Twentieth Entrant chooses to 'enter,' and the Chain
Store chooses to 'fight,' then the Chain Store's total payoff will increase by D. If, on the other
hand, the Chain Store chooses to 'acquiesce,' the Chain Store's payoff will increase by B. As B >
D, game theoretic rationality dictates that the Chain Store must choose 'acquiesce' if the
Twentieth Entrant 'enters.' Game theorists typically assumes that player rationality is Common
Knowledge:3 the Twentieth Entrant knows that the Chain Store is rational and will 'acquiesce.'
Hence, the Twentieth Entrant 'enters' because a payoff of B is better than the payoff of C, which
the Twentieth Entrant receives when it 'stays-out.' Due to Common Knowledge, the Nineteenth
Entrant knows the outcome of the twentieth stage and that it does not depend on the outcome of
the nineteenth stage. Following the above logic, if the Nineteenth Entrant 'enters,' the Chain
Store must 'acquiesce.' This logic is projected backward through the game, resulting in an 'enteracquiesce' outcome for all twenty stages. Selten named this type of outcome subgame perfect.
Because subgame perfection precludes non-credible threats, it has become a widely used solution
concept.
2.3 The Paradox
Selten calls the logic that identifies the subgame perfect solution 'Induction Theory.' He
contrasts this with 'Deterrent Theory.' To motivate deterrent theory, Selten considers the effect a
For a precise game theoretic definition of "Common Knowledge" see Drew Fudenberg and Jean Tirole. Game
Theory. MIT Press, 1996 page 542.
3
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"fighting" Chain Store would have on Entrants who a priori believed in induction theory.
Induction would lead the First Entrant to 'enter.' If the Chain Store 'fights' the First Entrant, this
would be a point of confusion for the other players. The remaining Entrants would struggle to
reconcile the Chain Store's action with their beliefs about Common Knowledge rationality, the
payoffs of the game, and the implications of subgame perfection. If the first several Entrants
entered and were all fought, Selten posits that some Entrants would change their beliefs and
assume that the Chain Store will 'fight' if they enter. These Entrants might then change their
strategy from 'enter' to 'stay-out.'
Selten acknowledged that in the final stage, it seems very unlikely that the Chain Store
would still fight, so induction should still hold at the end of the game. However, examine the
Chain Store that plans to 'acquiesce' in the last three rounds of the game, but 'fight' all Entrants in
the first seventeen rounds. If only seven of the first seventeen Entrants choose to 'stay-out,' this
"fighting" Chain Store's payoff is higher than that of a Chain Store which followed the strategy
dictated by inductive reasoning. Thus, deterrent strategy seems to be both a "rational" and an
appealing option for the Chain Store. Indeed, Selten observed that he never met anyone who
said that "he would behave according to induction theory [were he the Chain Store]."4 To Selten,
the disconnect between the game theoretic outcome dictated by induction theory and the
plausible outcome dictated by deterrent theory, constitutes a paradox.
2.4 Game Theory's Resolution
The dominant game theoretic resolution to the Chain Store Paradox hypothesizes that an
Entrant who saw a Chain Store fighting, might believe that there was a different type of Chain
Store for which such behavior was not irrational. As long as this fight-prone type of Chain Store
might exist, a normal Chain Store could fight to deter Entrants without violating the rules of
4
Selten (1978) pg 133
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game theory. Milgrom and Roberts (1982) added types by assuming that some percentage of the
Chain Stores had removed their ability to acquiesce when an entrant enters. Kreps and Wilson
(1982) assumed that there were two types of Chain Stores, one of which had payoffs that
encourage fighting. They justified this approach by claiming that it better captured the real
world story behind chain stores that fight price wars.
2.5 Paradox Unresolved
The current game theoretic resolution to the Chain Store Paradox alters the game in order
to avoid the conflict between deterrent logic and inductive logic. This does not resolve the
paradox; it merely avoids the paradox. Irrespective of whether one believes that the typing
solution captures the intricacies of a particular incumbent-entrant scenario, the existence of a
game (Selten's) in which game theory fails to explain human behavior calls into question
strategies reliance on game theory.
The typing approach is even less appealing when one considers publicly traded
companies, whose payoffs and motives are largely part of the public record. For example,
Microsoft's returns to a strategic move can be reasonably estimated, and it is unlikely that the
firm is driven by manager's psychic payoffs if those payoffs differ widely from the firm's bottom
line.
Even if one overlooks the fact that the orthodox game theoretic approach avoids the
paradox, this approach is still empirically inconsistent. To explain how typing enables rational
Chain Stores to play a deterrent strategy, both Kreps and Wilson, as well as Milgrom and
Roberts propose a specific Sequential Equilibrium (SE). However, in the limited human
experiments of the Chain Store game, the predictions of this SE failed (Jung, Kagel, and Levin
1994). Even if the SE is modified to account for the possible adoption of incorrect "homegrown"
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priors, as proposed in Camerer and Weigelt (1988), the SE solution concept is still unable to
organize the experimentally observed behavior.
3. EVOLUTIONARY COMPUTATION AS AN ALTERNATIVE SOLUTION CONCEPT
Most game theoretic analysis assumes that all agents adopt a carefully defined, selfinterested rationality. Game theory does not cast this pay-off maximizing rationality as the
dominant decision-making norm of business people from western cultures in the late 20th and
early 21st centuries. Rather, the predictive game theoretic canon requires that, within bounds, all
actors possess some degree of this homogeneous rationality.
Controlled human experiments poorly support such beliefs. Beyond the Jung et al.
experiments discussed above, other experimental results from developed western societies have
deviated from the general predictions of game theory (McKelvey and Palfrey 1992; Hoffman,
McCabe, and Smith 1996; McCabe and Smith 2000; Colman 2003). As noted above, there is
also a wealth of experimental data suggesting that culture and society have a large impact on the
outcome of these games (Henrich and Smith 2003, Hill and Gurvan 2003, Tracer 2003, and
Alvaed 2003).
The gap between the experimental findings above, and those forecast by predictive game
theory suggests a role for an alternative solution concept. The new concept should better fit the
observed data better and allow for cultural differences in behavior, but preserve the many
important and useful findings of orthodox game theory.
One candidate paradigm is evolutionary learning (Axelrod 1984; Axelrod 1987; Miller
1988; Holland and Miller 1991; Young 1993; Andreoni and Miller 1995; Young 1996; David
1999; Ünver 2001; Arifovic 2001; Hart 2002; Haruvy, Roth and Ünver 2006). This approach
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typically makes at least one of two claims. First, some claim that the underlying generating
mechanisms of evolutionary learning models better capture the processes by which groups obtain
decision heuristics. Second, some assert that such models can better predict real world behavior.
The results of evolutionary learning models typically confirm to the predictions of classical game
theory. However, some experiments (Andreoni and Miller 1995; Ünver 2001) suggest that
evolutionary computation is marginally better than game theory at predicting human behavior.
There are several approaches used to operationalize evolutionary learning, including
Evolutionary Algorithms (EAs). EAs evolve populations of agents. Each agent possesses an
individual rule system that converts inputs from the external environment into an action. The
model subjects these rule systems to evolutionary pressures such as selection and mutation.
These processes are similar to the decision heuristics used by actual economic agents. The
behavior of a particular model depends on the parameters and evolutionary rules used. By
searching through the parameter space, these models can also help bind the settings in which
certain classes of strategies evolve.
This paper's primary hypothesis contends that there exists a set of mechanisms and
parameters for an EA that yield a resolution to the Chain Store paradox. The implicit null
hypothesis asserts that all possible EAs replicate the predictions of classical game theory.
4. TECHNICSAL DETAILS: AN EA MODEL OF THE CHAIN STORE GAME
4.1 The Model
The Evolutionary Algorithms (EAs) used in this paper simultaneously co-evolve two
distinct agent populations: a population of Chain Stores (CSs) and a population of Entrants.
Each agent has a decision rule and a fitness score. In the first generation, the agents' decision
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rules (discussed below) are randomly generated. The decision rules populating later generations
are the product of a fitness-based evolution of the preceding generation. Specifically,
tournament selection was used to identify parents for the next generations. The more agents
entered into the tournament, the more elitist the selection mechanism.
To reduce the computational burden, this simulation examined a Chain Store game with
nine stages, instead of Selten's 20. Figure 2 shows the flow of the algorithm.
---------------------Insert Figure 2 about here
--------------------4.2 The Fitness Function
The EA's fitness function employs the joint-payoff correspondence used in Selten 1979.
In each generation, agents play multiple games. An agent's fitness at the end of the generation is
the sum of the agent's payoff from all stages in all games played during the current generation.
Table 2 correlates the outcomes of the ith stage of a game, to the change in the agents' fitness
scores.
---------------------Insert Table 2 about here
--------------------4.3 Decision Rules
Many different mechanisms could operationalize the agents' decision rules. This paper
employs two different structures. Both of these structures condition the agents' actions on the
current game's history. However, the first structure significantly collapses the histories,
circumscribing the agents' strategy space. The second structure allows the agents to evolve their
own collapsation scheme, and thus allows a much larger strategy space.
4.3.1 Binary String Decision Rule
The first decision rule mechanism is a simple binary string in which the CS agents treat
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all histories the same, and the Entrant's collapse all histories into one of two states. The first
state includes all histories congruent with a CS that fights all the time. The second state includes
all other histories.
The CS agents' binary string is a vector of nine binary elements;
cCS = (g1, g2, …, g9)
where gj
{1, 0}. Although the evolution system does not employ crossovers, the structure of
his system is similar to the Genetic Algorithms originally proposed in Holland (1975). In the
parlance of Genetic Algorithms, a bit's value is called an allele. The decision rule only queries
the Chain Store's string if an Entrant 'enters.' If the Entrant 'enters' in the ith stage, the decision
rule queries the ith bit of the Chain Store's string. If that bit's allele is '0' the Chain Store
'acquiesces.' If the ith bit's allele is '1', the Chain Store 'fights.'
The Entrant's string contains 17 bits;
cE1= (g1 g2, …, g17)
where gi
{1, 0}. At the start of each game, the decision rule queries the first bit. If that bit's
allele is '1' the Entrant stays out. If the allele is '0' the Entrant enters. There are two bits
associated with the each of the following eight stages; the first is queried if the CS has ever
acquiesced, the second is queried otherwise. Table 3 provides a summary of these decision
rules.5
---------------------Insert Table 3 about here
--------------------4.3.2 Binary String Parenting and Mutation
As shown in Figure 2, tournament selection is used to choose parents for the next
generation. At the starts of the tournament, X agents are chosen at random from a population.
5
Other structures were tested, such as limiting the Entrants’ memory. However, these structures did not produce
significantly different results.
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Of the X agents chosen, the agent with the highest fitness score parents an agent in the new
generation. The algorithm continues to run tournaments until the size of the new population
equals that of the old population. During the parenting process, each bit mutates with a
probability M. When a bit mutates, its allele is flipped. The values of X, M, and the size of the
populations were system parameters subjected to sensitivity analysis (discussed below);
however, the results reported in this paper set these parameters to 10, 2%, and 32 respectively.
4.3.3 Finite State Machine Based Decision Rules
The strategy space associated with the above BS decision rule is small; the CS agents
choose among 29 strategies and the Entrants choose among 217. This quality makes the system
easy to analyze. However, this also raises the possibility that the circumscribed strategy space
drives any observed behavior. It is therefore desirable to confirm the system's behavior using a
decision rule mechanism that allows for most, if not all, conceivable strategies. In theory, one
could extend the string so that agents could condition their response on every possible history.
This would require a string with 9841 bits for both the CS and the Entrant. Genetic Algorithms
with chromosomes that long are intractable. Such models become 'lost' in the solution space,
resulting in behavior that is too complex to be meaningfully analyzed.6
Evolving Finite State Machines (FSMs), rather than binary strings, avoids this problem
(Miller 1988; Holland and Miller 1991). In order for a binary sting to collapse histories, the
coder must predetermine the truncation scheme. However, evolutionary FSMs can evolve their
own truncation schemes. Therefore, evolutionary FSMs allow the possibility of very
complicated strategies, yet typically avoid getting 'lost' in the solution space.
6
The specification was tested for this paper and produced no describable results. Chen and Ni (1996) tried this
approach with a eight stage Chain Store game for which history was truncated to only include the opponents
behavior. However, they found that this abbreviated version was also intractable.
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The model in this paper uses a 16-state FSM to represent agents' decision rules. Each
state in the FSM contains an action and one transition pathway for each possible action the
agent's opponent can make during a stage of the game. The transition pathways frequently lead
to a new state, but can also return to its originating state. The agents always start in their
automaton's first state. Their opponent's actions govern the transition to subsequent states.
Both the Chain Store agents and the Entrant agents follow this general structure, but there
are some differences in the operationalization of these agents. In the Chain Store game, the
Entrant always moves first and the Chain Store only acts if the Entrant chooses to enter. The
Entrant agents always execute the action in their automaton's current state. Chain Store Agents
only execute the action in their automaton's current state if their opponent entered. Additionally,
each state in an Entrant's finite automaton has three transition pathways; one followed if the
Chain Store fights; one followed if the Chain Store acquiesces; and one followed if the Chain
Store didn't have the opportunity to act (because the Entrant didn't enter that round). The finite
automata operating the Chain Store agents only have two transition pathways; one followed if
the Entrant enters; and one followed if the entrant stays out. Figure 3 shows a possible three
state FSM for a CS, which instructs the agent to fight the first two entrants that enter, and
acquiesce thereafter.
---------------------Insert Figure 3 about here
--------------------4.3.4 Finite State Machine Parenting and Evolution
With the exception of the mutation function, the parenting and evolution processes used
with the FSM version are identical to those used in the BS version. In the FSM version, both the
actions and the destination of the transition pathways are susceptible to mutation. When a
transition pathway is stochastically selected for mutation, a new destination of the pathway is
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taken as a uniform random draw between one and 16. The method for mutating the actions
within each state of FSM is identical to that used to mutation the bit's alleles in the BS version of
the model
5: RESULTS AND DISCUSSION
5.1 Analytical Framework and Basic Behavior
With nine stages, the Chain Store game has 512 NE. To ease the analysis of the system's
behavior, we collapse all of the NE with an equal number of stages ending in 'stay-out.' This
reduces the 512 NE can be reduced to 10 states.7 Table 4 enumerates these ten states and shows
the number of NE associated with each. This collapsation scheme associates exactly one state
the Chain Store game's unique subgame perfect equilibrium, SGP. The other nine states are
defined by the number of stages that deviate from their SGP outcome (enter / acquiesce).
---------------------Insert Table 4 about here
--------------------Rather than converge to a particular state, mutation causes the system to orbit its basins
of attraction. It is difficult to determine the optimal size of these neighborhoods. This paper
heuristically defines the neighborhoods as having 70% convergence to a state. This convergence
is calculated at the stage level.8 This collapsation scheme is not part of the results, it is merely a
tool to make the results easier to describe.
Table 5 shows the percentage of time the system spends in each state. Each time step is
7
These observations are made at the phenotypical level. When an Entrant 'stays out' the Chain Store does not have
the opportunity to respond. Therefore, NE outcomes include all games in which the all stages end in either 'enteracquiesce' or 'stay-out.
8
For example, if a particular generation resulted in more than 70% of all stages ending in “enter-acquiesce,” the
system would be in the SGP state. If, more than 70% of stages 1-4 and stages 6-9 ended in “enter-acquiesce,” and
stage 5 ended in “stay-out” more than 70% of the time, the system would be in the SGP-1 state. If, more than 70%
of stages 1-4 and stages 6-9 ended in “enter-acquiesce,” and stage 5 ended in “stay-out” less than 70% of the time,
the system would not be in one of the ten states.
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one generation and the data in Table 5 were generated via a 20,000 generation simulation.
Although the heuristically defined neighborhoods occupy about 3% of the possible solution
space, the BS and FSM models respectively spend more than 80% and 66% of their time in a
neighborhood. The subgame perfect neighborhood is the most attractive neighborhood.
However, of the time spent in a neighborhood, both the BS and FSM versions spend about twice
as much time in non-SGP neighborhoods as in the SGP neighborhood. The key finding in these
results is that the system is about twice as likely to be orbiting a non-SGP NE as it is the SGP
NE.
---------------------Insert Table 5 about here
--------------------The second to bottom row, labeled "Transition," shows the amount of time the system
spent in between state neighborhoods. Table 5's bottom row, labeled "Fight-Trans," shows the
percentage of "Transition" time-steps in which at least one stage ended in "enter-fight" at least
70% of the time. The trade-off for the greater flexibility of the FSM models is, ceteris paribus, a
slower learning rate. This slower learning rate engenders longer periods of "off-equilibrium"
fighting.
We can treat the system's ergodic behavior as an approximate Markov Process.9 While
the neighborhood surrounding the SGP state is the most absorptive, other states exhibit a similar
magnitude of absorption.
5.2 An Example State Transitions: Genetic Drift and Deterrence Revisited
9
When the system transitions from one state’s neighborhood to another, it can, with very low probability, “passes
through” the neighborhood of a third state. The system only approximates Markov process, because knowledge that
the system has been in a state for several generations rules out the (almost trivially small) possibility that the system
is “passing through.” This technically violates the Markov property.
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To illustrate the mechanism behind the system's ergodic behavior, it is useful to consider
a simple example cycle from the BS version. This example, depicted in Figure 4, shows the role
of deterrence and genetic drift in the system's behavior.
---------------------Insert Figure 4 about here
--------------------At Step 1: The system is in the '1Stay-Out' state, according to the state definitions in
Table 4. The population has converged to a neighborhood around a pure strategy NE, in which
the Entrants 'stay out' in the first stages and 'enter' in the last eight stages. The CSs will 'fight'
any Entrant that 'enters' in the first round, but 'acquiesce' otherwise.
From Step 1 to Step 2: At Step 2, the CSs do not fight Entrants that enter in the first
stage. The payoff to a CS that adopts the Step 2 strategy is at least as good as that earned by a
CS that employs the Step 1 strategy; both earn the same payoff against a non-mutant Entrant. If
mutation causes an odd Entrant to 'enter' in the first stage, a CS employing the Step 2 strategy,
outperforms a peer employing the Step 1 strategy. Even if mutation did not give the Step 2
strategy an evolutionary advantage over the Step 1 strategy, genetic drift could still cause the CS
population to converge on the Step 2 strategy. Because all (non-mutant) Entrants 'stay-out' in the
first stage, at Step 2 the system exhibits the same phenotypical behavior observed in Step 1, is
still in the '1 Stay-Out' state.
From Step 2 to Step 3: Starting from Step 2, consider a mutation that causes an Entrant
to 'enter' in the first round. Because the CSs no longer 'fight' Entrants that 'enter' in the first
round, this mutant Entrant will outperform its peers. Selective evolutionary pressure enables the
mutation to dominate successive Entrant populations. This completes the transition from Step 2
to Step 3, and engenders a phenotypical transition from '1 Stay-Out' state to SGP state.
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At Step 3 - GENETIC DRIFT: In Step 3, the CSs always 'acquiesce,' so there is no
evolutionary pressure on the bits that control the Entrant's behavior if a CS 'fights.' For example,
with a per bit mutation rate of 0.02, the probability of a CS in the SGP state mutating to 'fight' in
the first five stages is 0.00032. Consider a mutation that causes an Entrant to 'stay-out' if it
observed non-acquiescent behavior in the first five stages (i.e. the first five stages end in either
'stay-out' or 'enter-fight'). The expected payoff earned by this mutant is less than its non-mutated
peers. However, because the mutated bit is so seldom queried, the mutant Entrants typically
perform as well as their non-mutated peers. Under a stochastic selection mechanism, such a
mutation can dominate the population (Wright 1931). In the theoretical biology literature, this
phenomenon is called genetic drift. In this simulation, genetic drift is akin to the Entrant
population forgetting how to respond to CSs that fight.
From Step 3 to Step 4: The simulation's small population size and highly selective
tournaments increase the probability that a small number of parents will seed each successive
generation. This creates population bottlenecks, which increase the likelihood of genetic drift.
Although genetic drift does occur in this simulation, the probability of drift is lower than that of
the transitions previously discussed in this example. For this reason, Step 3 is this example's
most absorptive step, and the SGP state is the system's most absorptive state. When genetic drift
causes the population of Entrants to 'stay-out' in the second round if they observe nonacquiescent behavior in the first round, the system transitions from Step 3 to Step 4.
From Step 4 to Step 1 – DETERRENCE: At Step 4, the systems phenotypical behavior is
identical to the behavior at Step 3. Therefore, the system is still in the SGP state. From Step 4 a
mutant CS can exploit the genetic drift in the Entrant. By fighting in the first round, the mutant
CS deters its opponent from entering in the second round. Using the payoffs Selten originally
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proposed this mutant CS would outperform its non-fighting peers. Selective evolutionary
pressure enables the mutation to dominate successive CS populations. At this point, the system's
behavior is identical to that prescribed by Selten's 'deterrent' theory. This 'deterrent' behavior is
not stable, but it drives the system's transition back to Step 1: through mutation and selection the
Entrant's learn that it is better to 'stay-out' in the first round, because if they enter, they'll be
fought. The Entrant's also learn that it is better to enter in the second round.
The types of transitions in this example capture the mechanisms behind all of the
system's transitions. However, in practice a typical cycle may involve more steps and a greater
number of states. For example, to transition from Step 3 to Step 4 the bit that controls the
Entrants' behavior when one stage of non-acquiesce must drift. The likelihood of such drift is
less than the likelihood of drift in the bit that controls the Entrants' behavior after several stages
of non-acquiescent behavior: CS mutation is more likely to induce one stage of non-acquiescent
behavior, increasing the probability that mutant Entrants are weeded from the population before
drift can occur. In practice, transitions typically involve CSs fighting for several stages to
exploit multiple instances of drift in latter stages.
5.3 Confirming the role of Genetic Drift
Genetic drift's role in enabling a system to move away from a subgame perfect NE has
been previously observed in evolution of simpler symmetric games (Harrald and Morrison 2001;
Arifovic 2001; Eaton and Morrison 2003). However, it has not been confirmed with agents this
complex.
In addition to inspection, two tests were used to confirm the role of drift in these
simulations. First, in the BS version transitions away from the SGP state should be positively
correlated with drift in the bits that determine the Entrants' actions in the non-acquiescent
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behavior. This effect was confirmed with a Probit regression where the dependent variable was
whether the system transitioned away from the SGP state in the next generation, and the
independent variables were the percentage on 'stay-out' alleles in bits 9 through 19. Table 6
displays the results of this regression.
---------------------Insert Table 6 about here
--------------------It is more difficult to run this type of statistical test on the FSM version of the model. A
second test was developed to collaborate the above evidence. In this second test, the mutation
mechanism was altered to prevent genetic drift. To turn genetic drift off in the BS version, only
bits that had been queried in the preceding generation were allowed to mutate. In the FSM
version, only states that had been queried in the preceding generation and only transition
pathways that had been activated in the preceding generation were allowed to mutate. As Table
7 shows, once the mutation scheme was changed to prevent genetic drift, the systems converged
to the SGP state, which effectively became an absorbing state. The FSM version of the model
did not converge as quickly as BS, because the FSM's genetic structure is more complex. The
bottom row in Table 7 shows percentage of stages that ended in the Entrant "staying out." This
statistic can be used to quickly compare the degree various trials deviate from convergence to the
SGP state.
---------------------Insert Table 7 about here
--------------------5.4 Number of Entrants
In Selten's version of the Chain Store game, one chain store faced a series of 20 entrants.
In their response to that paper, Kreps and Wilson (1982) note that "the analysis is unchanged in
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there is a single rival with repeated opportunities to enter."10 However, in this paper's models,
this equivalence did not hold.
This paper used two methods to operationalize multiple Entrants. The first co-evolving 9
distinct populations of agents, each population consisted of agents which played in a fixed stage.
In each game, each CS agent was paired with exactly one agent from each of the nine Entrant
populations. The first stage of the game was played between the CS and the Entrant from the
first population. As the game unfolded, all the Entrants "followed along" with the game.
However, the Entrants only acted in the stage matching the population was evolved to play. The
change in each Entrant's fitness function was solely determined by the outcome of the stage that
the Entrant played. This approach can also be understood as one potential entrant that allows it's
branch offices to independently search for their strategy.
The second method co-evolved only one population of Entrants. In each game, each CS
agent was paired with exactly nine agents from the Entrant population. Each agent played
exactly one stage of the game, but that stage was chosen at random. As the game unfolded, all
the Entrants "followed along" with the game and switched states appropriately. However, the
Entrants only acted in the stage they were chosen to play. The change in each Entrant's fitness
function was solely determined by the outcome of the stage that the Entrant played. In this
version, it was convenient for the number of agents to be a multiple of nine. Rather than the
standard 32 agents per population, this version evolved 36 agents per population. Because there
were an equal number of CS agents and Entrant agents, each Entrant played exactly one stage in
exactly nine times as many games as did each CS agent.
10
Kreps and Wilson (1982) pg. 254
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Table 8 displays the results from the multi-entrant trials for both the BS and FSM
versions of the model. Trials in which multiple Entrant populations were evolved deviated much
less from the SGP state than did the original specification. The behavior from trials that evolved
two populations, but had nine different Entrants in each game fell between the two extremes.
---------------------Insert Table 8 about here
--------------------Given the role of genetic drift in moving the system away from the SGP state, the results
in Table 8 suggest that trials with multiple Entrants should be less susceptible to genetic drift
than were the original versions. In BS version, the dominant form of drift involves the bits that
handle the Entrants behavior in the face of non-acquiescent behavior. Specifically, when the
system is in the SGP state, these bits become neutral, whereupon drift causes them to shift from
'enter' to 'stay-out.' The survival rate of the 'stay-out' allele in these bits can be used as a proxy
for genetic drift; the lower the survival rate of the 'stay-out' allele, the lower the likelihood of
drift. Where the survival rate is the number of parents with an allele, divided by the number of
times the allele appeared in the population. Table 9 compares the average survival rates of the
'stay-out' allele in bit 10: the bit that controls the Entrants' behavior in the face of nonacquiescent behavior in the first stage. The data in Table 9 support the hypothesis that
specifications with a multiple Entrants are more susceptible to drift.
---------------------Insert Table 9 about here
--------------------There are two drivers behind the decreased susceptibility to drift. In the 9-population, 9Entrant trails, each population only optimizes behavior in one stage. This focus increases the
efficacy with which less fit alleles are weeded from the population. However, this does not
explain the difference between the 2-population, 9-Entrant trails and the original 2-populaiton, 2-
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entrant models. In the 2-population, 2-Entant versions, the ability to weed out week alleles is
further limited by the difficulty is separating the impact of a response to a 'bad' event from the
event itself.
Consider an example starting from the dominant SGP (always-enter, always-acquiesce)
state. A mutation can cause a CS to fight entrants in round X. From the Entrant's perspective, an
allele that causes it to respond by 'staying-out' in round Y (Y>X) is evolutionarily inferior. The
occasional mutation on behalf of the CS can provide the Entrant population(s) with the
opportunity to weed out weak alleles; thus, lowering the probability of a genetic drift induced
transition to a non-SGP state. In the 2-population, 9-Entrant version, an Entrant that 'enters' in
stage Y (after observing a CS fighting in stage X) is more fit than a peer that "stays-out" against
such a CS, and as fit as a peer that did not encounter a "fight in stage X" CS. In the 2population, 9-Entrant version, an Entrant that 'enters' in stage Y (after observing a CS fighting in
stage X) is more fit than a peer that "stays-out" against such a CS, but less fit than a peer that
did not encounter a mutant CS. Under a selective tournament system, this limits the degree to
which occasional mutations ward-off genetic drift in the 2-population, 2-Entrant versions.
5.6 Parameters and Sensitivity Analysis
Unless otherwise noted all the results displayed were from trials with the following
parametric settings;
•
32 agents per population
•
10 agents per selection tournament
•
9 stages per game
•
3 games per generation
•
2% probability of mutating a bit in the BS models
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•
5% probability of mutating a string in the FSM models
•
20,000 generations per trial
ANTs algorithms (Miller 1998) were used to search the parameter space. The genetic
drift phenomenon only occurs over a small area of the mechanism and parameter spaces. The
most important element of the parameter space is mutation. If mutation is extremely low, the
agents learn slowly and frequently get stuck in non-NE ruts. The likelihood of getting stuck
increased as the number of games per generation decreased. With low mutation and one game
per generation, the models were slow to switch stages.
The optimal mutation rate for the observation of genetic drift is about a 15% chance that
each agent is mutates at least once.11 Significant increases in the mutation rate minimized the
probability of genetic drift, but greatly increased the background noise associated with the
system's behavior.12
The selectivity, measured by the number of agents in each tournament, has some impact,
but this impact is minimal. For example, decreasing the number of agents from 10 to 5 in the
standard FSM version only lowers the "average stay-out" measure from 28.1% to 27.0%.
The long number of generations per trial is useful for the FSM versions, which typically
take several hundred to a few thousand generations to settle into their normal behavior patterns.
The BA versions of the model settle into their normal behavior patterns quite quickly.
11
To balance mutation rates across agents with different lengths of chromosome material, mutation was
implemented on a per-bit basis. The per-bit probability of mutation was equal for the CS and Entrant agents.
However, this led to difference in the probability that a parent agent was mutated at least once. In the BS version,
the per-bit probability of mutation was 2%. The probability of at least one mutation was 16.6 and 30.5, respectively
for the CS and Entrant agents. In the FSM the probability of mutation was 5% per 16-bit string. The probability of
at least one mutation was 14.3 and 18.5, respectively for the CS and Entrant agents.
12
Similar results were observed with a simple model where agent behavior was not based on histories; i.e. each
agent had a nine-bit bit. At moderate levels of mutation, this system fully collapsed to the SGP state. At moderate
levels of mutation, the system occupied a large neighborbood around the SGP state. At high levels of mutation the
system did not converge.
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6 DISCUSSION AND CONCLUSION
6.1 Implications for Strategic Analysis and the Chain Store Paradox
The role of genetic drift in this system's behavior offers an important insight into strategy.
Strategies that are not tested can degenerate over time, and competitors can exploit this
degeneration. This type of degeneration is seen in both the knowledge of a strategy, and the
ability to implement a strategy. For example, firms that seldom face crises must rely on an
external crisis-management firm when a crisis arises. Because the time between crises is so long,
the firm forgets how to handle a crisis. However, firms that frequently face crises typically
handle the situation themselves. An example of strategic capacity degeneration can be seen in the
US automobile industry. Although the big three developed the capacity to engineer increasingly
fuel efficient cars in the early 1980s, this capability degenerated during the cheep-oil / big-SUV
era in 1990s and early 2000s. This capacity degeneration is currently being exploited by
Japanese auto-makers, which spent the 1990s and early 2000s improving their ability to engineer
increasingly efficient cars. This paper's findings also suggest that decentralizing decision
making can damping the likelihood that a firm suffers from strategic drift. As Table 8 shows,
when the potential entrant conducts a local strategy search in each market, the drift phenomenon
is less prevalent.
The behavior induced by genetic drift also offers a potential resolution to the Chain Store
Paradox. Selten asserts that, "The fact that the logical inescapability of induction theory fails to
destroy the plausibility of the deterrent theory is a serious phenomenon which merits the name of
a paradox."13 The ergodic Markov process produced by this paper's evolutionary model
incorporates the logical inescapability of induction theory: the unique subgame perfect an
13
Selten (1978) pg 133 .
.
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equilibrium predicted by induction theory remains the most absorptive state in the Markov
process. However, the system's Markov chain also reflects the plausibility of the deterrent
theory. A process, similar to deterrence enables the system to transition from the SGP state to
other NE.
The Markov chain solution implies that the game's current equilibrium depends on its
history. This explains the behavior of "off-the-street" players. When a random player
approaches the game, she does not know the game's specific history. As a result, she cannot
know the games current equilibrium. The player is likely to assume her opponent also does not
know the game's current equilibrium. Therefore, a rational Chain Store player might use
deterrence to convince her opponent that they are at an equilibrium that is advantageous to the
Chain Store. This explains the prevalence of deterrent behavior that Selten deemed paradoxical.
The paradox's real driver was classical game theory's insistence that these games have a
singular deterministic solution derived from a universally applied, homogeneous solution
concept. Within an evolutionary computational framework, both deterrent and induction theories
drive the behavior of the ergodic system that 'solves' the Chain Store game. That one theory
does not destroy the other need not constitute a paradox.
This finding applies to real world market entry decisions. Consider Microsoft's
dominant, near monopolistic, position in several product markets. One might posit that Microsoft
would fight a potential entrant, because Microsoft believes that it can drive the entrant from the
market. However, one might argue that an entrant which incurs the start-up costs needed to
challenge Microsoft would be unwilling to exit the short or medium term. Under this latter
assumption classical game theory and this papers model offer radically different strategic
assessments. Because Microsoft is a publicly traded firm the cost of its fighting a price war can
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be reasonably well estimated. Additionally, the transparent management structure of a publicly
traded firm limits the likelihood that a product manager could fight an unprofitable price war
simply because she gets a physic benefit from fighting. Therefore, the "typing" solutions
proposed by classical game theory do not apply and "enter-acquiesce" is the only possible
outcome. However, this paper's model suggests that a price war is possible despite Microsoft's
public status.
In the last year, there have been two entrants in market dominated by Microsoft.
Microsoft's monopoly in online, consol-based gaming (Xbox Live) was challenged by Sony's
2006 PlayStation Network launch. A few months before the PlayStation Network launch, Google
launched their Google Apps product line, which directly challenged Microsoft's Microsoft Office
Suite. In both cases, the entrants preemptively avoided a price war by pricing these products at
zero for existing customers. For Google, this move is in line with their zero-pricing strategy.
However, this represented a strategic shift for Sony. Sony's actions are in line with the
implications of the evolutionary model presented in this paper.
6.2 Implications for Evolutionary Computation
This paper shows that decision paradigm predicated on strategic co-evolution can account
for the influence of both deterrent and induction theories in the Chain Store paradox. Hence, this
paper's primary hypothesis is accepted and the null hypothesis is rejected. This adds supports the
claim that social scientists can fruitfully apply co-evolutionary computational models to strategic
analysis. This does not imply that evolutionary mechanisms fully explain strategic interaction.
Rather, this paper's findings simply support the inclusion of evolutionary computation in
strategic analysis.
Much of the analytic work in this field considers evolutionary mechanisms that do not
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encourage genetic drift. In particular, the mechanisms considered in Canning (1992), Kandori,
Mailath, and Rob (1993), Young (1993), and Young (1996) operationalize mistakes (or
mutation) by a random draw from the complete strategy space, rather than a one-bit permutation
to strategies currently active in the population. While these mechanisms are analytically
tractable, they are unlikely to produce genetic drift. The widespread use of drift-dampening
mechanisms has led to a pervasive view, summarized by Young (1993);
[In cases where the Markov process representing the system has no pure
absorbing state,] "it has a stationary distribution that describes the relative
frequency with which different states are observed in the long run … if the
probability of mistakes [or mutation] is small, then this stationary
distribution is concentrated around a particular subset of pure strategy
Nash equilibria. In fact, typically it puts almost all weight on exactly one
equilibrium. This stochastically stable equilibrium [original italics] will
be observed with a probability close to one when the noise is very
small."14
The experiments in this paper support the above view when genetic drift is turned off. In
these cases, the Chain Store game's unique subgame perfect NE satisfies Young's
characterization of a dominant stochastically stable equilibrium. However, when the mutation
mechanism promotes genetic drift, a singular stochastically stable equilibrium is less likely to
dominate the system's behavior. These findings are inline with computationally derived findings
in Arifovic (2001), Harrald and Morrison (2001), and Eaton and Morrison (2003).
Whether a modeler should use a drift promoting agent structure and evolution mechanism
14
P. Young 1993, pgs 59-60
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Paradox Lost
depends entirely on the phenomena under consideration and the goals of the model. However, it
is important to realize that genetic drift can impact a model's behavior when choosing the
structure and mechanisms for an evolutionary model.
6.3 Limitations and Future Work
This paper's computational model matched the results of Jung et al.'s human trials in two
ways. First, both showed a deviation from the subgame perfect solution in trials without
multiple chain store types. Second, both showed a larger deviation from the subgame perfect
solution in trials with multiple chain store types.
Most of the results in Jung et. al focused on how individuals learned to behave after
playing the game at least 30 times. However, the study of iterative individual learning is an area
to which evolutionary computation is ill suited. Evolutionary computation models the dynamics
of iterative group learning. This circumscribes the degree to which evolutionary computation
can organize data from Jung et al.'s human experiments.
The demonstrated efficacy of evolutionary computation in strategic analysis suggests a
need for the increased use of data driven evolutionary models. In particular, "Friedman-esque"
instrumentalist mechanism selection and parametric specification should enable data driven,
evolutionary, computational models to make significant contributions to strategy, game theory,
mechanism design, and financial analyses.
FIGURES AND TABLES:
Enter
Chain Store
Fight
Acquiesces
D, D
B, B
Stay-out
C, A
TABLE 2: STAGE i'S ∆ FITNESS
Entrant
Entrant
TABLE 1: STAGE PAYOFFS
Enter
Chain Store
Fight
Acquiesces
+0, +0
+2, +2
Stay-out
+1, +5
FIGURE 1: EXTENSIVE FORM OF A TWO-STAGE CHAIN STORE GAME
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Paradox Lost
Note: from top to bottom, payoffs listed are for: Chain Store; Entrant 1; Entrant 2.
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FIGURE 2: FLOW OF THE EVOLUTIONARY ALGORITHM
STEP ONE: Each CS is randomly paired with an Entrant
STEP TWO: For each pair, the Entrant’s decision rule determines
whether the Entrant ‘enters’ or ‘stays-out’ of the first stage. If
the Entrant ‘enters,’ the CS’s decision rule determines whether
the CS ‘fights’ or ‘acquiesces.’ This step is repeated sequentially
for the game’s nine stages. The decision rules may be
conditioned on the outcomes of the current game’s earlier stages.
STEP THREE: After each stage the agents’ fitness scores are
updated following Selten’s original specification (see Table 2)
STEP FOUR: If the agents have played less than X games during
the current generation, the algorithm returns to step one. When
the agents have played exactly X games, the algorithm precedes
to step five.
STEP FIVE: The current generation ‘parents’ a new generation. Z
agents are randomly chosen, with replacement. The agent with
the highest fitness score is selected as a parent; its decision rule is
copied into the new generation, but may be subjected to
mutation. This process is repeated, until the new generation’s CS
and Entrant populations equal the size of the old populations.
STEP SIX: If the simulation has evolved less than G generations,
the algorithm returns to step one. When the simulation has
evolved exactly G generations, the algorithm terminates.
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TABLE 3: BS VERSION STRING STRUCTURE
String
Bit
Bit
String
Bit
Bit
FIGURE 3: SAMPLE THREE-STATE CS-FSM
Enter
Stay Out
Fight
Fight
Stay Out
(2)
(1)
Enter
Enter
Acquiesc
e
(3)
TABLE 4: STATES FOR THE CHAIN STORE GAME
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Stay Out
Paradox Lost
State Name
SGP
1 Stay-Out
2 Stay-Out
3 Stay-Out
4 Stay-Out
5 Stay-Out
6 Stay-Out
7 Stay-Out
8 Stay-Out
9 Stay-Out
# of stages ending in
enter-acquiesce
Don't enter-fight
9
8
7
6
5
4
3
2
1
0
0
1
2
3
4
5
6
7
8
9
# of NE
1
9
36
84
126
126
84
36
9
1
TABLE 5: % OF TIME IN STATE
SGP
1 Stay-Out
2 Stay-Out
3 Stay-Out
4 Stay-Out
5 Stay-Out
6 Stay-Out
7 Stay-Out
8 Stay-Out
9 Stay-Out
Transition
Fight-Trans
BS
FSM
28.6
24.4
13.8
6.3
3.3
2.3
1.5
0.5
0.2
0.1
19.2
50.5
17.9
4.4
6.1
11.0
13.7
10.0
2.5
0.1
0.2
0.1
33.9
83.2
FIGURE 4: EXAMPLE OF TYPICAL CYCLE FOR BS VERSION
STEP 1:
State = 1 Stay-Out
Entrants ‘stay out’ in the first stage and
‘enter’ in the other stages, irrespective of
Chain Store behavior.
Chain Stores ‘fight’ mutant Entrants that
enter in first stage and ‘acquiesce’ in the
other stages.
STEP 4:
State = SGP
STEP 2:
State = 1 Stay-Out
If the Chain Store fights in the first stage,
the Entrant ‘stays out’ in the second stage.
Otherwise the Entrant always ‘enters’
Entrants ‘stay out’ in the first stage and
‘enter’ in the other stages, irrespective of
Chain Store behavior.
Chain Stores always ‘acquiesces’
Chain Stores ‘acquiesce’ whenever an
Entrant enters
STEP 3:
State = SGP
Entrants ‘enter’ all stages, irrespective of
Chain Store behavior.
Chain Stores always ‘acquiesces’
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TABLE 6: PROBIT - THE LIKELIHOOD OF TRANSITIONING AWAY FROM THE SGP STATE REGRESSED ON
THE % OF 0'S IN BS VERSION'S ENTRANTS' 'DETERRENT' GENES
Note: gene 18 dropped due to collinearity
Number of obs =
6649
LR chi2(8)
=
84.31
Prob > chi2
=
0.000
Log likelihood = -1003.7431
Pseudo R2
=
0.040
Coefficient Std. Error
z
P>z
[95% Conf. Interval]
constant
-1.978
0.040
-49.970 0.000
-2.055
-1.900
% of 0's in Gene 10
0.954
0.201
4.750
0.000
0.560
1.348
% of 0's in Gene 11
0.427
0.107
3.990
0.000
0.217
0.637
% of 0's in Gene 12
0.412
0.096
4.310
0.000
0.225
0.600
% of 0's in Gene 13
0.230
0.093
2.470
0.013
0.048
0.413
% of 0's in Gene 14
0.238
0.089
2.670
0.008
0.063
0.412
% of 0's in Gene 15
0.215
0.088
2.450
0.014
0.043
0.387
% of 0's in Gene 16
0.272
0.094
2.880
0.004
0.087
0.457
% of 0's in Gene 17
0.046
0.107
0.430
0.667
-0.164
0.256
TABLE 7: % OF TIME PER STATE - DRIFT VS NO DRIFT
SGP
1 Stay-Out
2 Stay-Out
3 Stay-Out
4 Stay-Out
5 Stay-Out
6 Stay-Out
7 Stay-Out
8 Stay-Out
9 Stay-Out
Transition
Fight-Trans
Average
Stay-out
Original
BS
Original
FSM
No Drift
BS
No Drift
FSM
28.6
24.4
13.8
6.3
3.3
2.3
1.5
0.5
0.2
0.1
19.2
50.5
17.9
4.4
6.1
11.0
13.7
10.0
2.5
0.1
0.2
0.1
33.9
83.2
99.7
0.1
0.1
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.2
23.5
95.2
0.0
0.0
0.3
1.5
0.0
0.2
0.0
0.0
0.0
2.6
47.1
16.0
28.1
0.0
1.1
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Paradox Lost
TABLE 8: % OF TIME PER STATE - MULTIPLE ENTRANTS
Original
BS
2 Pop -9
Entrant
BS
9 Pop. 9 Entrant
BS
Original
FSM
2 Pop. 9 Entrant
FSM
9 Pop. 9 Entrant
FSM
SGP
1 Stay-Out
2 Stay-Out
3 Stay-Out
4 Stay-Out
5 Stay-Out
6 Stay-Out
7 Stay-Out
8 Stay-Out
9 Stay-Out
Transition
Fight-Trans
28.6
24.4
13.8
6.3
3.3
2.3
1.5
0.5
0.2
0.1
19.2
50.5
66.3
9.2
5.7
2.3
1.1
0.5
0.3
0.2
0.0
0.0
14.5
47.8
85.1
6.6
2.7
0.9
0.1
0.0
0.0
0.0
0.0
0.0
4.5
26.7
17.9
4.4
6.1
11.0
13.7
10.0
2.5
0.1
0.2
0.1
33.9
83.2
44.5
4.7
0.8
2.5
5.6
2.9
1.3
0.7
1.4
2.5
33.3
80.6
23.4
29.2
12.1
2.7
0.8
0.1
0.1
0.0
0.0
0.0
31.6
9.0
Avg. Fight
16.0
7.0
2.0
28.1
18.2
12.3
TABLE 9: BIT 10 SURVIVAL RATE
Bit 10 Survival
Rate
Original BS
2 Pop. – 9
Entrant BS
9 Pop. – 9
Entrant BS
0.990
0.960
0.917
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